Hans-Walter Lorenz

Nonlinear dynamical economics and chaotic motion

The behavior of nonlinear dynamical systems can differ completely from that of linear dynamical systems. Regular oscillatory motion as well as seemingly irregular motion can emerge in these systems without resort to exogenous influences or peculiar parameter values. Recent mathematical advances made in the theory of nonlinear dynamical systems allow the restrictive linear approach to dynamical phenomena in economics to be discarded and observable fluctuations of economic variables to be modelled in a simple and convenient way. This book attempts to familiarize the reader with the standard tools in nonlinear dynamical systems theory. The usage of these tools is demonstrated with simple examples from different fields like business cycle theory, optimal control theory, growth theory, and population dynamics. The presentation encompasses a short reminder of linear dynamical systems and traditional themes in nonlinear systems like the existence and uniqueness of limit cycles and closed orbits in predator-prey systems. The main part of the book deals with chaotic motion in economic systems. It is demonstrated, that irregular dynamical behavior can be generated in deterministic economic systems with relative ease, once the linear approach in modelling dynamic phenomena is abandoned. One-dimensional and higher-dimensional difference equations as well as higher-dimensional differential equation systems are discussed and illustrated with a variety of economic examples. Recently emerging techniques for discriminating stochastic and deterministic dynamical systems are presented, as well as the question of whether the nature of actual economic time series is chaotic or stochastic.

Strange attractors in a multisector business cycle model

Abstract Three coupled oscillating sectors in a multisector Kaldor-type business cycle model can give rise to the occurrence of chaotic motion. If the sectors are linked by investment demand interdependencies, this coupling can be interpreted as a perturbation of a motion on a three-dimensional torus. A theorem by Newhouse, Ruelle and Takens implies that such a perturbation may possess a strange attractor with the consequence that the flow of the perturbed system may become irregular or chaotic. A numerical investigation reveals that parameter values can be found which indeed lead to chaotic trajectories in this cycle model.

International trade and the possible occurrence of chaos

Abstract The paper demonstrates that international trade among fluctuating economies may involve the occurrence of a strange attractor and hence of chaos. Once autonomous economies display cyclical behavior, international trade activities can be considered as a perturbation of the motion of the autonomous economies, possible implying chaos via a theorem by Newhouse, Ruelle and Takens (1979).

On the role of expectations in a dynamic Keynesian macroeconomic model

Abstract A simple Keynesian macroeconomic disequilibrium model with rationing is considered. This paper investigates the influence of different expectations hypotheses (regarding next periods goods prices) on the models dynamic behavior when intertemporal substitution effects exist. While the bifurcation behavior is not qualitatively influenced when expectations are formed according to adaptive expectations or an unweighted-averages hypothesis, the dynamic behavior of the model can be changed in an essential way when pattern-recognition expectations are assumed. So-called perfect cyclic expectations can turn a previously chaotically moving economy into a system with a low-periodic motion in particular cases.

On the complexity of simultaneous price-quantity adjustment processes

While the Walrasian price tâtonnement represents the traditional dynamic process in the general equilibrium context with and without production, Walras and other classics designed the process exclusively for pure exchange economies. In productive economies, the short-run output adjustment of existing firms and the entry/exit of firms should be modeled as well. So-called cross-dual processes which represent the classical approach to the dynamics of productive economies are discussed and extended. Complex motion can emerge in a discrete-time version of the original two-dimensional system when the aggregate demand function has a non-standard shape. A simultaneous process of price and short-run quantity adjustment with free entry and exit of competitive firms in a single market with a continuum of firms can generate closed orbits via a Hopf bifurcation when the slope of the demand function is positive at equilibrium. When the continuum economy is replaced by an economy with a finite number of firms, noisy limit cycles and complicated behavior can be observed.

Complexity in Deterministic, Nonlinear Business-Cycle Models — Foundations, Empirical Evidence, and Predictability

Business-cycle theory represents one of the oldest fields in economics. While it was treated as a nearly esoteric field in specialized graduate texts during the late 1960s and early 1970s, the last fifteen years saw it resurrecting even as a synonym for dynamic macroeconomics. The Rational Expectations literature of the late 1970s and early 1980s and the development of sophisticated econometric tools in investigations of an economy’s fluctuations occasionally seemed to encourage the believe that business-cycle theory was an invention of the so-called New Classical economics. However, it is a fact that the observed cycling of an economy constituted the major impetus for many a classical and neoclassical economist in the 19th and early 20th century to engage in economic theorizing at all. Haberler’s (1937) seminal text on the history of business-cycle theory demonstrates in an enlightening fashion that the ups and downs in economic activity were central not only in - to name just a few - Hawtrey’s (1913), Hayek’s (1933), Marx’s (1867), Pigou’s (1929), or Sismondi’s (1837) work but that numerous, usually forgotten writers concentrated on oscillations in particular markets or the entire economy.

Complexity in Economic Theory and Real Economic Life

Complex dynamic behaviour in terms of chaotic motion, catastrophic events or other seemingly irregular and unexpected features of and in theoretical economic models – aimed at describing real-world phenomena – are nowadays known as a common property of many nonlinear approaches to an understanding of the motion of actual time series, such as inflation rates, unemployment figures, and many other – mainly macroeconomic – economic variables. Since most existing models in economic dynamics are constructed in the tradition of classical mechanics, this result does not appear as a real surprise. However, the real ‘complexity challenge’ for economic theory still persists in identifying the complex structure of economic reality, which cannot be satisfactorily represented by simple deterministic laws of motion, although such ‘laws’ might possess the possibility of very complicated dynamic motion.

Bifurcation Theory in Dynamical Economics

This chapter deals with a subject that has become a major focus of research in dynamical economics during the last decade, namely bifurcation theory. Central to this topic is the question whether the qualitative properties of a dynamical system change when one or more of the exogenous parameters are changing. In contrast to the physical sciences, it is usually impossible to assign a definitive, once-and-for-all valid number to most parameters occurring in dynamical systems in economics. Parameters are introduced into an economic model in order to reflect the influence of exogenous forces which are either beyond the scope of pure economic explanation or which are intentionally considered as being exogenously given from the point of view of partial theorizing. It is desirable to determine whether the qualitative behavior of a dynamical system persists under variations in the parameter space. Thus, the results of bifurcation theory are especially important to dynamic modelling in economics.

Economics, Chaos and Environmental Complexity

Every scientist who is aware of the possible emergence of chaotic motion in nonlinear dynamical systems should be familiar with the idea that longer-term weather predictions might be impossible. E.N. Lorenz’ work on a three-dimensional dynamical system arising in the context of a climate model is almost always cited as the starting point of modern research into nonlinear dynamical systems. The so-called butterfly effect even serves as a vehicle for demonstrating unexpected and dramatic consequences of the presence of non-linearities in dynamical systems. However, for a non-specialist in this field, the theoretical reasons for the emergence of chaotic motion are usually unclear (if one is interested in qualitative justifications of the investigated mathematical nonlinear dynamical systems). Paul Higgins provides fascinating insights into theoretical approaches to the weather and climate phenomena in his paper and concentrates on the idea that the interaction of subsystems (like atmos phere, ocean, biosphere etc.) is responsible for the observable weather and cli mate phenomena. This global system consisting of interacting subsystems (which reminds of a so-called coupled oscillator system in dynamical systems theory which is known to exhibit chaotic motion when the dimension of the system is sufficiently high) implies the existence of multiple equilibria, with an important consequence.

Nichtlineare Dynamik in der Ökonomie

Die dynamische Wirtschaftstheorie beschaftigt sich mit der modellhaften Beschreibung von Veranderungen wirtschaftlicher Variablen im Zeitablauf. Als „Modell“ wird in diesem Zusammenhang ein mathematisch beschreibbares Gerust von Verhaltensweisen und technischen Zusammenhangen bezeichnet, welches letztlich dynamische Systeme in Form von Differenzen- oder Differentialgleichungssystemen definiert. Die Veranderung volkswirtschaftlicher Variablen wie z.B. des Bruttosozialprodukts, der Inflationsrate oder der Arbeitslosenquote und betriebswirtschaftlicher Grosen wie dem Unternehmensumsatz oder der Verweildauer in Produktionswarteschlangen werden mit diesem Ansatz grundsatzlich als das Ergebnis der Interaktion wirtschaftlicher Einheiten angesehen, bei denen in verschiedenster Form Verknupfungen zwischen aufeinanderfolgenden Zeitpunkten zu verzeichnen sind. Die entstehenden dynamischen Systeme konnen dabei ausschlieslich deterministischer Natur sein; in vielen dynamischen okonomischen Betrachtungen wird jedoch angenommen, das die Entwicklung wirtschaftlicher Variablen auch von weiteren, exogen bestimmten, sowohl okonomischen als auch nicht-okonomischen Grosen beeinflust wird.